In the following, we pinpoint the cause of the subtle difference in solution concepts between the baseline Bayesian Stackelberg model and our proposed model. Considering the geometric properties of \(\widehat{\Pi }\)  and the piece-wise linearity of \(V^\mu \) , we adopt a mathematical programming viewpoint. Moreover, the geometric intuition, in fact, leads to a finite-time algorithm for searching the \(\epsilon -\) Stackelberg equilibrium. Similar to our partition \(\mathcal {P}^\gamma \) , in the baseline Bayesian Stackelberg model, each follower's pure strategy also corresponds to a convex polytope region in the space mixed strategy, and when restricting the leader's utility to an arbitrary polytope, the maximum value is achieved at some extreme point of the polytope. This is not surprising as seeking the optimal leader's strategy is equivalent to solving LPs, and naturally, the solution is some vertex of the feasible domain, and we refer the reader to [1]} for more details and to [2]} for a visualization.
